Importance and reduction of the sidewall-induced band-broadening effect in pressure-driven microfabricated columns

The influence of the detailed design of the sidewall region upon the over-all band-broadening in microfabricated packed-bed or collocated monolithic support structure (COMOSS) columns has been investigated using computational fluid dynamics (CFD) simulation techniques. It is shown that, under unretained solute conditions, very small structural variations of the order of only 5% of the particle diameter can give rise to a 4-fold increase of the band-broadening. A comprehensive study has been made to quantify this effect as a function of the fluid velocity, the particle diameter, the channel widths, and of course, the sidewall region design. Because the sidewall effect can be fully attributed to a mismatch between the flow rates in the column center and in the sidewall region, it is fortunately also quite straightforward to avoid it. A very simple design, yielding band-broadening values identical to that of a hypothetical sidewall-less column for all possible values of the flow velocity, the particle diameter, or the channel width is proposed.     

A kinetic study of mass transfer in reversed-phase liquid chromatography on a C18-silica gel

The characteristic features of mass-transfer kinetics in a reversed-phase (RP) column packed with a C18-silica were studied. The relevant information on phase equilibrium thermodynamics and mass-transfer kinetics was obtained by frontal analysis and the pulse method, respectively. The equilibrium isotherm was accounted for by the simple Langmuir model. The ratio of the axial dispersion coefficient to the mobile-phase flow velocity increased almost linearly with increasing solute concentration. Similarly, the mass-transfer rate coefficient (km) showed a linear dependence on the solute concentration. The positive concentration dependence of km resulted from that of the surface diffusion coefficient, which was interpreted with the chemical potential driving force model. The contribution of axial dispersion to band broadening was predominant in the RP column packed with the medium-size packing material used (particle diameter, 12 microns) whereas that of the kinetics of adsorption/desorption was negligibly small. The results of this study demonstrate how an analysis of the dependence of the mass-transfer kinetics on the flow velocity and the solute concentration allows a better understanding of this kinetics.     

A molecular dynamics approach in simulations of fluid-solid particles flow

In the paper a discrete system of particles carried by fluid is considered in a planar motion. The volumetric density of particles is assumed to be small enough such that they can be treated within the framework of a molecular dynamics model. The fluid is then considered as a carrier of particles. The Landau-Lifshitz concept of turbulence is used to describe the fluctuating part of fluid velocity. This approach is applied to simulate different regimes (laminar and turbulent) and various states of particle motion (moving bed, heterogeneous flow, and homogeneous flow) using only two parameters, which have to be determined experimentally. These two parameters, found for a particular pipe and for a particular velocity from a simple experiment, then can be used for other pipe diameters and different velocities. The computer simulations performed for the flow of particles in pipes at different flow velocities and different pipe diameters agree favorably with experimental observations of the type of flow and critical velocities identifying transitions from one type to another. Received: 8 January 1999     

WALL DEPOSITION OF SMALL SUSPENDED PARTICLES IN A TURBULENT CHANNEL FLOW

The diffusion of small suspended particles in a turbulent channel flow is studied by solving the transport advection-diffusion equation. The mean flowfield in the channel is simulated using a two-equation k-ε turbulence model. Deposition velocity is evaluated at different sections in the channel for different particle sizes and flow Reynolds numbers. The effects of turbulence dispersion and Brownian diffusion on particle deposition velocity are discussed. The variation of particle deposition velocity with particle diameter, density and flow Reynolds number are analyzed. The wall deposition velocities for different size particles are compared with those obtained by other models.     

Motion of rigid aggregates under different flow conditions

The response of rigid aggregates to different flow fields is investigated theoretically using model clusters with realistic three-dimensional structure composed of identical spherical primary particles. The aim is to relate the main fluid dynamic properties of the system with the geometry and morphology of the aggregates. Our simulations are based on Stokesian dynamics. The dilute limit of a colloidal aggregate system was studied, where aggregates are very far from each other, and hence, mutual inter-aggregate interactions are negligible. The motion of aggregates is characterized in terms of translational mobility and angular velocity, and the ability of simple models, based on either the simplified aggregate geometry or the concept of permeability, to capture the main features of the motion is examined.     

On diffusion theory in turbulence

The Fokker-Planck equation for the probability density of fluid particle position in inhomogeneous unsteady turbulent flow is derived. The equation is obtained starting from the general kinematic relationship between velocity and displacement of a fluid particle and applying exact asymptotic analysis. For (almost) incompressible flow the equation reduces to the convection diffusion equation and the equation pertaining to the scalar gradient hypothesis. In this way the connection is established with eddy diffusivity models, widely used in numerical codes of computational fluid dynamics. It is further shown that within the accuracy of the approximation scheme of the diffusion limit, diffusion constants can equally be based on coarse-grained Lagrangian statistics as defined by Kolmogorov or on Eulerian statistics in a frame that moves with the mean Eulerian velocity as proposed by Burgers. The results presented for diffusion theory are the leading terms of asymptotic expansions. Truncated terms are higher-order spatial derivatives of the probability density or of the scalar mean value with coefficients based on cumulants higher than second order of fluid velocities and their derivatives. The magnitude of these terms has been assessed by employing scaling rules of turbulent flows in pipes and channels, turbulent boundary layers, turbulent jets, wakes and mixing layers, grid turbulence, convective layers and canopy turbulence. It reveals that a true diffusion limit does not exist. Although truncated terms can be of limited magnitude, a limit process by which these terms become vanishingly small and by which the diffusion approximation would become exact does not occur for any of the cases of turbulent flow considered. Applying the concepts of diffusion theory resorts to employing approximate methods of analysis.     

Dispersion reduction in open-channel liquid electrochromatographic columns via pressure-driven back flow

Application of electrokinetic forces to drive the mobile phase diminishes analyte dispersion in open-channel liquid chromatographic columns due to minimization of shear in the flow field. However, the retentive layer coating the inner walls of such devices slows down the average convective velocity of solute molecules in its vicinity, inherently causing dispersion of analyte bands. In this article, we explore the possibility of reducing such dispersion in electrochromatographic columns by imposing a pressure-driven back flow in the system. Analysis shows that although such a strategy introduces shear in the flow field, the overall dispersion in the mobile phase is reduced. This occurs as the streamline velocity in such a system is greater near the channel walls than that in the center of the conduit, thereby allowing fluid dispersion to counteract wall retention effects. For an optimally chosen magnitude of the back flow, hydrodynamic dispersion of any target species in the mobile phase may be shown to diminish by a factor of 3 and 10/3 in a circular tube and a parallel-plate geometry, respectively. A similar reduction in slug dispersion is also realized in rectangular conduits for all aspect ratios. In trapezoidal geometries with large wedge angles or isotropically etched profiles, this reduction factor may attain values of 10 or greater.     

Collective drag and sedimentation: comparison of simulation and experiment in two and three dimensions

We simulate systems of particles immersed in fluid at Reynolds numbers on the particle scale of 0.1 to 20. Our simulation method is based on a finite differencing multi-grid Navier-Stokes solver for the fluid and a molecular dynamics technique for the particle motion. The mismatch between the fixed rectangular grid and the spherical particle shape is taken into account by considering analytical series expansions of the pressure and velocity of the fluid in the vicinity of the particle surface. We give an expression for the force on a particle in terms of the expansion coefficients. At each time step these coefficients are determined from pressure and velocity values on the fluid grid. We demonstrate the validity of our approach by performing numerical simulations of flow through porous solid beds and of bulk sedimentation in two and three spatial dimensions. We compare our results to experimental data and analytical results. Quantitative agreement is found in situations where the volume fraction remains below approximately 0.25 both in two and three dimensions, provided that at the same time the Reynolds number remains below about 10. In contrast, e.g., to finite-element techniques the method remains fast enough to allow dynamical simulations of particle-fluid systems with several hundred spheres on workstations taking all inertial effects into account.     

A wall boundary treatment using analytical volume integrations in a particle method

Numerical treatment of complicated wall geometry has been one of the most important challenges in particle methods for computational fluid dynamics. In this study, a novel wall boundary treatment using analytical volume integrations has been developed for two-dimensional (2D) incompressible flow simulations with the moving particle semi-implicit method. In our approach, wall geometry is represented by a set of line segments in 2D space. Thus, arbitrary-shaped boundaries can easily be handled without auxiliary boundary particles. The wall's contributions to the spatial derivatives as well as the particle number density are formulated based on volume integrations over the solid domain. These volume integrations are analytically solved. Therefore, it does not entail an expensive calculation cost nor compromise accuracy. Numerical simulations have been carried out for several test cases including the plane Poiseuille flow, a hydrostatic pressure problem with complicated shape, a high viscous flow driven by a rotating screw, a free-surface flow driven by a rotating cylinder and a dam break in a tank with a wedge. The results obtained using the proposed method agreed well with analytical solutions, experimental observations or calculation results obtained using finite volume method (FVM), which confirms that the proposed wall boundary treatment is accurate and robust.     

Revisiting dynamics and models of microsegregation during polycrystalline solidification of binary alloy

Microsegregation formed during solidification is of great importance to material properties.The conventional Lever rule and Scheil equation are widely used to predict solute segregation.However,these models always fail to predict the exact solute concentration at a high solid fraction because of theoretical assumptions.Here,the dynamics of microsegregation during polycrystalline solidification of refined Al-Cu alloy is studied via two-and three-dimensional quantitative phase-field simulations.Simulations with different grain refinement level,cooling rate,and solid diffusion coefficient demonstrate that solute segregation at the end of solidification (i.e.when the solid fraction is close to unit) is not strongly correlated to the grain morphology and back diffusion.These independences are in accordance with the Scheil equation which only relates to the solid fraction,but the model predicts a much higher liquid concentration than simulations.Accordingly,based on the quantitative phase-field simulations,a new analytical microsegregation model is derived.Unlike the Scheil equation or the Lever rule that respectively overestimates or underestimates the liquid concentration,the present model predicts the liquid concentration in a pretty good agreement with phase-field simulations,particularly at the late solidification stage.     

Numerical solution of the complete mass balance equation in chromatography

A new approach to simulate the movement of bands through a chromatographic column is presented. Similar to the Craig distribution model, the mass balance equation is divided into two equations describing two successive processes. The first equation includes two effects: solute diffusion in the mobile phase and migration of the solute band with the mobile phase. The second equation deals with the distribution of the solute between phases, i.e., the adsorption isotherm. The partial differential equations are integrated numerically over time and space using two methods. The first approach is a finite difference method. In the second approach, the propagation operator is expanded in a Chebyshev series, where large time steps can be used. The rate of adsorption and desorption is determined by the size of the time increment. By varying the size of the time step, it is possible to study kinetic effects. The influences of sample size, injection width, rate of mass transfer, and mobile phase velocity on the elution profile were studied. Simulations using the modified Craig approach with either of the two numerical procedures showed that the solutes behaved in the chromatographically expected manner. Moreover, with linear adsorption isotherms, direct relationships between HETP, as well as retention times, and experimental parameters could be established.     

Direct numerical simulation of particulate flows with heat transfer in a rotating cylindrical cavity

The purpose of this work was the direct numerical simulation of heat and fluid flow by granular mixing in a horizontal rotating kiln. To model particle behaviour and the heat and fluid flow in the drum, we solve the mass conservation, momentum and energy conservation equations directly on a fixed Eulerian grid for the whole domain including particles. At the same time the particle dynamics and their collisions are solved on a Lagrangian grid for each particle. To calculate the heat transfer inside the particles we use two models: the first is the direct solution of the energy conservation equation in the Lagrangian and Eulerian space, and the second is our so-called linear model that assumes homogeneous distribution of the temperature inside each particle. Numerical simulations showed that, if the thermal diffusivity of the gas phase significantly exceeds the same parameter of the particles, the linear model overpredicts the heating rate of the particles. The influence of the particle size and the angular velocity of the drum on the heating rates of particles is studied and discussed.     

Numerical investigation of hydrodynamics in liquid-solid circulating fluidized beds under different operating conditions

A computational fluid dynamics (CFD) model was employed to investigate the hydrodynamics of the liquid–solid circulating fluidized bed (LSCFB). The numerical simulations of the flow in the LSCFB under different operating conditions, including different superficial liquid velocities, superficial solid velocities, particle densities and shapes, are carried out using the CFD model developed in the previous work. The numerical predictions show correct trends and good agreements with the experimental data. It is demonstrated that the radial non-uniformity and axial uniformity exist in the flow structures under different operating conditions. By increasing the superficial solid velocity, the average solids holdup and radial non-uniformity increase, while the opposite trends are observed by increasing the superficial liquid velocity. Besides, the solids holdup decreases with the decrease in the particle density. It is also observed that all the flow distributions in the radial and axial directions in LSCFBs are more uniform than those in GSCFBs.     

Particle migration of concentrated suspension flow in bifurcating channels

Flow of suspension in bifurcating channels has extensive applications in industrial and natural settings. A phenomenon of particular interest during the flow of concentrated suspension is shear-induced particle migration. Previous works on suspension transport in branched channels have been limited to dilute flow conditions. We have carried out experiments using the Particle Image Velocimetry (PIV) technique to study concentrated suspension transport in asymmetric T- and symmetric Y-shape channels. Numerical simulations of fluid flow and particle transport equations were also carried out for the same geometry which was used in the experiments. The migration and transport of particles in the simulations were studied using the Diffusive Flux Model. We have observed in both experiments and numerical simulations that due to the shear-induced migration phenomena the particles move towards the center of the channel, and this gives rise to the blunting of velocity profile before the junction. After the bifurcation, the peak of velocity profile moves in the direction of the outer wall, whereas, the maxima in particle concentration was observed near the inner walls. This causes asymmetry in the velocity and concentration profiles in the daughter branches. As we move towards the downstream positions the maxima in velocity and concentration profiles again shifts toward the center of the channel. The results from the experiments and simulations are observed to be in good agreement.     

Effect of dialyzer reprocessing on glucose homeostasis

While dialyzer manufacturers only provide information about their products as a black box, this study aimed at optimizing dialyzer geometry by looking in detail at transport processes and fluid properties inside the dialyzer using numerical modeling. A three‐dimensional computer model of a single hollow fiber with its surrounding membrane and dialysate compartment was developed. Different equations govern blood and dialysate flow (Navier‐Stokes), radial filtration flow (Darcy), and solute transport (convection‐diffusion). Blood was modeled as a non‐ Newtonian fluid with a viscosity varying in radial and axial direction because of the influence of local hematocrit, diameter of the capillaries, and local shear rate. Dialysate flow was assumed as an incompressible, laminar Newtonian flow with a constant viscosity. The permeability characteristics of the asymmetrical polysulphone membrane were calculated from laboratory tests for forward and backfiltration. The influence of the oncotic pressure induced by the plasma proteins was implemented as well as the reduction of the overall permeability caused by the adhesion of a protein layer on the membrane. Urea (MW60) was used as a marker to simulate small molecule removal, while middle molecule transport was modeled using vitamin B12 (MW1355) and inulin (MW5200). The corresponding diffusion coefficients were determined by counting for the fluid and membrane characteristics. Fiber diameter and length were changed in a wide range for evaluation of solute removal efficiency. The presented model allowed us to investigate the impact of flow, hematocrit, and capillary dimensions on the presence and localization of backfiltration. Furthermore, mass transfer was found enhanced for increased fiber lengths and/or smaller diameters, most pronounced for the middle molecules compared to urea.     

Resolution of cross-type optical particle separation

A real-time, continuous optical particle separation method, termed cross-type optical particle separation, is investigated theoretically and experimentally. The trajectory of a particle subject to cross-type optical particle separation is predicted by solving the particle dynamic equation and compared with experimental data. For various flow velocities and particle sizes, the retention distances are measured where the displacement perpendicular to the fluid flow direction is referred to as the retention distance. The measured retention distances are in good agreement with theoretical predictions. The retention distance increases as the particle size increases due to the radiation force, but decreases as the flow velocity increases since the residence time of a particle in the laser beam decreases with increasing flow velocity. To evaluate the performance of the cross-type optical particle separation method, size-based separation resolution is derived theoretically in terms of the refractive index of the particle and instrumental fluctuations. Furthermore, an expression for the maximum resolution is derived.     

Enhancement of air purification by unique W-plate structure in two-stage electrostatic precipitator: A novel design for efficient capture of fine particles

In this paper, a novel W-plate two-stage ESP was developed and investigated systematically through the experimental and simulated process. Numerical models and available calculation procedure of solving coupling electrostatic field, fluid field, and particle dynamics were established, whose accuracy was validated by experiments. The relationship among collection efficiency, gas velocity, and particle diameter was studied, and the distribution of electrostatic field, the evolution of EHD flow and fluid field, and particle dynamics, including particle charging, particle trajectory, transverse velocity, and particle concentration, were also investigated thoroughly. Results showed that W-plate two-stage ESP exhibited excellent number-based collection efficiency for fine particles which benefited from the reasonable structure design and the exceeding weak influence of EHD flow. Besides, the particle charging process suggested that the diameter decided the dominant charging mechanism, and the trajectory also played an important role in controlling the charging action. Compared with the behavior of each particle injected at different inlet positions, fine particles injected near the discharge wire got more charging number and quicker capture. Importantly, W-plate structure could exert its crucial role in capturing particles with the help of fluid field and inertial effect when inlet gas velocity increased rapidly. W-plate two-stage ESP had more than 90% number-based collection efficiency for >3 μm diameter particles and more than 75% number-based collection efficiency for 0.3–1 μm diameter submicron particles at 2 m/s gas velocity in both experimental and simulated investigations.     

Simulation of dendritic growth in the platform region of single crystal superalloy turbine blades

A cellular automaton model of the solid-liquid interface, combined with a finite difference computation of solute diffusion has been developed to simulate single crystal solidification in molds with step changes in geometry. Simulations were carried out for columnar dendrites passing from the narrow airfoil region of a blade into the platform region, which has an increased cross-sectional area. Different shapes of isotherms moving at a constant velocity were considered in the simulations. The change in mold geometry leads to a significant increase in the undercooling in front of the dendrite tips as they spread around the mold corner. The model was applied to geometries investigated by prior authors, correctly predicting the formation of a <001> to <010> boundary observed experimentally.     

Dispersion in large aspect ratio microchannels for open-channel liquid chromatography

Solute dispersion in open-channel liquid chromatography is often dominated by transverse diffusion limitations in the mobile phase (Martin, M.; Guiochon, G. Anal. Chem. 1984, 56, 614-620) convecting the solute species. While such dispersion is known to scale with the square of the Peclet number based on the narrower dimension of the conduit, the proportionality constant may significantly vary with the aspect ratio of the channel geometry. In this article, we investigate the effect of channel sidewalls on axial dispersion in electrokinetically and pressure-driven chromatographic systems. The analysis presented here clearly identifies the contribution from flow, wall retention, and the interaction between the two to the overall slug dispersion in the mobile phase for any arbitrary channel geometry. The particular geometries that have been investigated in this work, however, are the rectangular and the isotropically etched profiles often employed in microanalysis systems. Further, the effectiveness of simple double-etched profiles proposed elsewhere (Dutta, D.; Leighton, D. T. Anal. Chem. 2001, 73, 504-513) to diminish the effect of channel sidewalls on Taylor-Aris dispersion has also been examined. Analysis shows that dispersion arising due to shear and wall retention, as well as the interaction between the two, may be significantly reduced in large aspect ratio microchannels for optimized channel geometries.